Externality

Consider an economy where there is a fishing company, F, and a steel company, S, both of which are

both are price setters. Steel production pollutes nature, which harms the fishing industry's profits. The cost function for S is: C_s(s, x) = 3s² + 2(s x)²

where s 0 is steel produced and x 0 is the amount of pollution. The cost function for F is: C_f(f, x) = f² + xf

where f 0 is output from fishing. Let the prices of steel be p_S = 18 and the price of fish be p_F = 7.

1. find the equilibrium(s*, x*, f*) and and corporate profits when it's free on polluters.
2. Find the social optimal level of pollution. (x**)
3. Now a tax on pollution is introduced tx for t 0. Find the tax level that ensures it socially optimal level of pollution x**
4. The company Waterworks arises and they're also negatively affected by the pollution in the form of increased costs. Without setting up any calculations. Can you say something about whether the tax should raised or lowered ensure the socially optimal level for all three companies?